2006-7 Quantum Theory Slides Lecture 2

Quantum theory and atomic spectroscopy Lecture 2 Discrete energy levels

The story so far…. • Atoms contain small charged particles • Spectroscopy reveals atoms absorb energy at discrete wavelengths • An element is defined by the number of protons in its nucleus • Now read on …

Periodic table again

• Ordering depends on number of protons • But why this arrangement? Why is the 2nd row longer than the 1st etc?

A couple of loose ends…… • How are electrons arrange- why don’t atoms collapse? • What do the lines in an atomic spectra actually mean?

Which element to pick?

• Notice that hydrogen, though lightest element, doesn’t have the “simplest” optical spectrum

Balmer and hydrogen ⎛ 1 1 ⎞ ν = R H ⎜⎜ 2 − 2 ⎟⎟ ⎝ n1 n 2 ⎠

• Balmer chose H since it was lightest element • Found simple expression to explain visible spectrum • Introduces the 1st quantum number n

Light bulbs and Planck

• Classical model is a black-body absorber • But classical model leads to a logical absurdity! • Quantum model- black-body contains quantised oscillators

Planck’s constant

E = hν

• A direct relationship between energy and frequency • Planck’s constant is tiny (6.634 x 10-34 Js) • We find it crops up everywhere in quantum theory.

Basic atomic model • Only quantized levels in an atom • Transitions between each level lead to discrete spectra

-19 11eV = 1.60218 x 10 eV = 1.60218 x 10-19JJ

c = νλ How are electrons arranged in an atom?

Two-level atom - 1

Lowest energy state = ground state

• The two level atom is a simple system for analysing problems in real atoms • The electron will have different energies in the two levels

What happens when we have MORE than one electron? Where does the electron go?

Two level atom - 2 Two electrons in two levels leads to four distinct configurations Classically- three different total energy states

Quantum – four different total energy states (SUPERPOSITION of c2 + c3)

Moseley’s X-rays • Increased charge means tighter bonding • Shift in x-rays

Photoelectron spectroscopy 1 • High-frequency radiation can ionise an atom, excess energy taken as electron kinetic energy.

1 hν = I.E. + m e v e2 2

Photoelectron spectroscopy - 2 • Use modified Balmer formula…. and find ionisation energy of hydrogenic atoms. ν IP

⎞ ⎛ 1 ⎛ 1 RH 1⎞ ⎟⎟ = R H ⎜⎜ 2 − 0 ⎟⎟ = = RH ⎜⎜ 2 − 2 ∞ n n n 1 ⎠ ⎝ 1 ⎠ ⎝ 1

ν IP = RH

ν IP = Z 2 R H

What happens if you have more than one electron?

Electron arrangement • The lack of a smooth change in ionisation energy indicates complex arrangement of electrons

Photoelectron spectra • Two ways to do this • 1) Fix light energy and record electron kinetic energies • 2) scan frequency of light and record electrons of fixed kinetic energy

Photoelectron spectra - 2 • This is the photoelectron spectrum of boron • Evidence for shells and subshells

New quantum number!! • The subshell structure can be explained using a new quantum number l, the orbital angular momentum quantum number • So if n = 1, l = 0 we have a 1s electron if n = 2, l = 0 we have a 2s electron if n = 2, l = 1 we have a 2p electron • The maximum number of electrons in a subshell is given by 2 x (2l + 1) • Allowed values of l: l = 0, 1…… (n-1)

Hydrogen energy levels • Hydrogen is unique- has degenerate subshells • The number of electrons in a subshell depends on l • s subshell holds 2 p subshell holds 6 d subshell holds 10 • Different shells have different numbers of subshells, hence different numbers of electrons

Answers to the earlier questions.. • Optical spectra due to transitions between energy levels • Electrons stop losing energy because there are no empty quantum energy levels left!

Hang on…… • Why do different subshells hold different numbers of electrons? • Next time, we will see that ANGULAR MOMENTUM is the key!